Compare the pair: Rotated versus unrotated surface codes at equal logical error rates

Practical quantum computers will require resource-efficient error-correcting codes. The rotated surface code uses approximately half the number of qubits as the unrotated surface code to create a logical qubit with the same error-correcting distance. However, instead of distance, a more useful qubit-saving metric would be based on logical error rates. In this work we find the well-below-threshold scaling of logical to physical error rates under circuit-level noise for both codes at high odd and even distances and then compare the number of qubits used by each code to achieve equal logical error rates. We perform Monte Carlo sampling of memory experiment circuits with all valid CNOT orders using the stabilizer simulator Stim and the uncorrelated minimum-weight perfect matching decoder PyMatching 2. We find that the rotated code uses about <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mrow><a:mn>74</a:mn><a:mo>%</a:mo></a:mrow></a:math> the number of qubits used by the unrotated code to achieve a logical error rate of <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mrow><b:msub><b:mi>p</b:mi><b:mi>L</b:mi></b:msub><b:mo>=</b:mo><b:msup><b:mn>10</b:mn><b:mrow><b:mo>−</b:mo><b:mn>12</b:mn></b:mrow></b:msup></b:mrow></b:math> at the operational physical error rate of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"><c:mrow><c:mi>p</c:mi><c:mo>=</c:mo><c:msup><c:mn>10</c:mn><c:mrow><c:mo>−</c:mo><c:mn>3</c:mn></c:mrow></c:msup></c:mrow></c:math>. The ratio remains <d:math xmlns:d="http://www.w3.org/1998/Math/MathML"><d:mrow><d:mo>≈</d:mo><d:mn>75</d:mn><d:mo>%</d:mo></d:mrow></d:math> for <e:math xmlns:e="http://www.w3.org/1998/Math/MathML"><e:mi>p</e:mi></e:math> values within a factor of two of <f:math xmlns:f="http://www.w3.org/1998/Math/MathML"><f:mrow><f:mi>p</f:mi><f:mo>=</f:mo><f:msup><f:mn>10</f:mn><f:mrow><f:mo>−</f:mo><f:mn>3</f:mn></f:mrow></f:msup></f:mrow></f:math> for all useful <g:math xmlns:g="http://www.w3.org/1998/Math/MathML"><g:msub><g:mi>p</g:mi><g:mi>L</g:mi></g:msub></g:math>. Our work finds the low-<h:math xmlns:h="http://www.w3.org/1998/Math/MathML"><h:msub><h:mi>p</h:mi><h:mi>L</h:mi></h:msub></h:math> scaling of the surface code and clarifies the qubit savings provided by the rotated surface code, providing numerical justification for its use in future implementations of the surface code.